MA103: Multivariable Calculus

Unit 2: Partial DifferentiationFor single-variable functions, the derivative of the graph at a
specified point is the slope of the tangent line at that point. This is
because the dependent variable is only changing with regards to a single
independent variable. In this course, we will consider functions of
multiple variables. Because these functions have outputs that depend on
multiple variables, each variable contributes to the rate of change of
the function independently. Thus, we refer to the function’s rate of
change with respect to a single specified variable as a partial
derivative, as it does not describe the entire rate of change of the
function. Moreover, for functions of multiple variables, there will no
longer be a single tangent line at a specified point. For example, in
the case of a function of two variables, we consider a tangent plane as
opposed to a tangent line. The concept of a partial derivative is
important to study concepts such as tangent spaces, extrema, etc. in the
context of functions of 2 or more variables.

As in Single-Variable Calculus, Multivariable Calculus studies the
maximum and minimum values for given functions due to their numerous
applications. Lagrange Multipliers is a method of finding maximum and
minimum values for functions subject to constraints that uses partial
derivatives; however, the idea of optimization with constraints has no
one-variable analogue.

Again, you should focus on the geometric meaning of the concepts
introduced in this unit.

Unit 2 Time Advisory
This unit will take you approximately 25.5 hours to complete.

☐ Subunit 2.1: 9.25 hours

☐ Subunit 2.1.1: 1.75 hour

☐ Subunit 2.1.2: 1.25 hour

☐ Subunit 2.1.3: 1.25 hour

☐ Subunit 2.1.4: 5 hours

☐ Subunit 2.2: 16.25 hours

☐ Subunit 2.2.1: 4.5 hours

☐ Subunit 2.2.2: 3.25 hours

☐ Subunit 2.2.3: 4.75 hours

☐ Subunit 2.2.4: 3.75 hours

Unit2 Learning Outcomes
Upon successful completion of this unit, the student will be able to:

Determine the domain of a function.

Determine whether the domain of a function is open or closed,
bounded or unbounded, connected or not connected.

Sketch the graph of the domain of a function.

Sketch the graph of a given function.

Identify the characteristics of a function from a graph of its level
curves.

Evaluate limits.

Show that limits do not exist by evaluating the limit along two
different paths.

Determine whether a given function is continuous.

Evaluate partial derivatives.

Evaluate higher order partial derivatives.

Describe the geometrical significance of a directional derivative.

Find the directional derivative.

Describe the relation between existence of partial derivatives,
continuity, and differentiability.

Evaluate derivatives of functions using the chain rule.

Use the method of Lagrange Multipliers to find the extrema of
functions subject to constraints.

Also Available in:
[HTML and Java](http://mathinsight.org/level_sets)</span>
NOTE: In order to view the Java applets within this resource you
must click on the [HTML and
Java](http://mathinsight.org/level_sets) link, as the PDF version
does not support the Java applets.
<span style="background-color: transparent;"> Instructions: Please
click on the webpage linked above and work through the notes and the
applets. Feel free to go through more examples by clicking on the
link at the bottom of the page. </span>
This activity should take approximately 1 hour to complete.
Terms of Use: The linked resource above is released under
a [Creative Commons Attribution-NonCommercial-ShareAlike 3.0
Unported
License](http://creativecommons.org/licenses/by-nc-sa/3.0/), it is
attributed to Duane Q. Nykamp and the original version can be
found [here](http://mathinsight.org/level_sets).

Instructions: Please view the entire lecture (52:18). You may also
click on the “Transcript” tab on the page to read the lecture.

This video should take approximately 1 hour and 15 minutes to watch
and review.
Terms of Use: The video above is released under [Creative Commons
Attribution-NonCommercial-ShareAlike
3.0](http://creativecommons.org/licenses/by-nc-sa/3.0/). It is
attributed to Denis Auroux and the original version can be found
[here](http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/video-lectures/lecture-10-second-derivative-test/).

Instructions: Please click on each link above to sections “2.1
Functions of 2 Variables,” “2.2 Limits and Continuity,” and “2.3
Partial Derivatives.” Read all four parts for each individual
section. Begin by reading the first webpage for each section linked
above, and then select the other Parts at the top of each webpage to
continue reading (After finishing Part 4 of "2.3 Partial
Derivatives", you will have read 12 Parts total).

This reading should take approximately 1 hour and 30 minutes to
complete.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

Instructions: Please click on the “Functions of 2 Variables
Exercises” link above, and at the top of this webpage, click on the
“Exercises” link to access the question sets. Work through
exercises 7, 9, 15, 19, and 25. Similarly, click on the “Limits and
Continuity Exercises” link above, and once on the webpage, select
the “Exercises” link at the top of the webpage to access the
question sets. Complete exercises 5, 13, 17, and 25. Finally,
click on the “Partial Derivatives Exercises” link above, and at the
top of this webpage, click on the “Exercises” link to be redirected
to the question sets. Try to solve exercises 5, 11, 19, 25, and
29. Check the solutions by clicking on the link titled “Answers to
Selected Odd
Exercises” (PDF)
on the main page, choosing chapter 2, and then finding sections
2.1-2.3.

These exercises should take approximately 3 hours to complete.

Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

*NOTE: In order to view the Java applets within this resource you
must click on the HTML and
Java link,
as the PDF version does not support the Java applets.

Instructions: Click
on the webpage linked above and work through the notes and the
applets. Feel free to work on more examples or read more sections by
clicking the relevant links on the bottom of the page.

Also available in: [Adobe Flash, iTunes, or
Mp4](http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/video-lectures/lecture-11-chain-rule/)
Instructions: Please view the entire lecture (50:09). You may also
click on the “Transcript” tab on the page to read the lecture.
Terms of Use: The video above is released under [Creative Commons
Attribution-NonCommercial-ShareAlike
3.0](http://creativecommons.org/licenses/by-nc-sa/3.0/). It is
attributed to Denis Auroux and the original version can be found
[here](http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/video-lectures/lecture-11-chain-rule/).

Instructions: Please click the links titled “Section 2.5
Linearization and the Hessian” and “Section 2.6 The Chain Rule”
above. For each section, read Parts 1-4 in their entirety. You may
access each part by clicking on the links at the top of each
webpage.

These exercises should take approximately 3 hours to complete.

Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

Instructions: Please click the link titled “2.5 Linearization and
the Hessian” posted above, and then select the “Exercises” link at
the top of the webpage to access the questions. Try to solve
exercises 1, 9, 21, and 27. Then, click on the “2.6 The Chain Rule”
link posted above, and select the “Exercises” link at the upper
right corner of the webpage to access the questions. Complete
exercises 5, 13, and 23. Check the solutions by clicking on the
link titled “Answers to Selected Odd
Exercises” (PDF)
on the main page, choosing chapter 2, and then scrolling down to
find sections 2.5 and 2.6.

These exercises should take approximately 2 hours to complete.

Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

Instructions: Please view entire video lecture (50:10). You may
also click on the “Transcript” tab on the page to read the
lecture.
This video should take approximately 1 hour and 15 minutes to watch
and review.
Terms of Use: The video above is released under [Creative Commons
Attribution-NonCommercial-ShareAlike
3.0](http://creativecommons.org/licenses/by-nc-sa/3.0/). It is
attributed to Denis Auroux and the original version can be found
[here](http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/video-lectures/lecture-12-gradient/).

Instructions: Please read all four parts of “2.7 Properties of the
Gradients.” Begin by reading “Part 1: Gradients and Level Curves”
linked above, and then select the links for Parts 2-4 at the top of
the webpage to continue reading.

This reading should take approximately 30 minutes to read and
review.

Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

Instructions: Please click on the link above titled “2.7 Properties
of the Gradients Exercises.” On this webpage, click on the link
titled “Exercises” at the upper right corner of the webpage to
access the questions. Try to complete exercises 3, 13, and 15.
Check the solutions by clicking on the link titled “Answers to
Selected Odd
Exercises” (PDF)
on the main page, choosing chapter 2, and scrolling down to section
2.7.

These exercises should take approximately 1 hour to complete.

Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

*NOTE: In order to view the Java applets within this resource you
must click on the HTML and
Java link,
as the PDF version does not support the Java applets.

Instructions: Click
on the webpage linked above and work through the notes and the
applets. Feel free to work on more examples or read more sections by
clicking the relevant links on the bottom of the page.

Instructions: Please view the entire video lecture (50:10). You
may also click on the “Transcript” tab on the page to read the
lecture.

This video should take approximately 1 hour and 15 minutes to watch
and review.
Terms of Use: The video above is released under [Creative Commons
Attribution-NonCommercial-ShareAlike
3.0](http://creativecommons.org/licenses/by-nc-sa/3.0/). It is
attributed to Denis Auroux and the original version can be found
[here](http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/video-lectures/lecture-13-lagrange-multipliers/).

Instructions: Please click the links titled “Section 2.8
Optimization” and “Section 2.9 Lagrange Multipliers” found above.
Read Parts 1-4 of each section. Make sure to click on the links to
each part at the top of each webpage to complete this reading
assignment.

This reading should take approximately 1 hour to read and review.

Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

Instructions: Please click on the link above titled “2.8
Optimization Exercises”, and then select the “Exercises” link at the
top right corner of the webpage to access the question sets. Solve
exercises 1, 17, and 25. Similarly, click on the link titled “2.9
Lagrange Multipliers Exercises” above, and then click on the link
titled “Exercises” at the upper right corner of the webpage to
access the questions. Work on exercises 5, 7, and 15. Check the
solutions by clicking on the link titled “Answers to Selected Odd
Exercises” (PDF)
on the main page, choosing chapter 2, and scrolling down to sections
2.8 and 2.9.

These exercises should take approximately 2 hours to complete.

Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

Instructions: Please view the video lectures in their entirety
(49:11 and 45:23, respectively). You may also click on the
“Transcript” tab on the page to read the lecture.
Terms of Use: The video above is released under [Creative Commons
Attribution-NonCommercial-ShareAlike
3.0](http://creativecommons.org/licenses/by-nc-sa/3.0/). It is
attributed to Denis Auroux and the original version can be found
[here](http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/video-lectures/lecture-14-non-independent-variables/)
(Lecture 14) and
[here](http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/video-lectures/lecture-15-partial-differential-equations/)
(Lecture 15).

Instructions: Please read Parts 1-4 of “Section 2.4: Partial
Differential Equations” from the link posted above. Begin by
reading “Part 1: Partial Differential Equations” linked above, and
then click on the links to Parts 2-4 at the top of the webpage.

This reading should take approximately 30 minutes to study.

Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

Instructions: Please click on the “2.4: Partial Differential
Equations Exercises” link above, and once on the webpage, click on
the “Exercises” link at the top right corner of the webpage to
redirect to the exercise sets. Complete exercises 11, 21, and 23.
Check the solutions by clicking on the link titled “Answers to
Selected Odd
Exercises” (PDF)
on the main page, choosing chapter 2, and scrolling down to section
2.4.

These exercises should take approximately 1 hour to complete.

Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.